By using the theoretical concepts presented, the engineering students have to develop the ability to analyze a given problem in a logical manner. More in details, the students should know how to analyze the equilibrium of a statically determined plane system of rigid bodies, by obtaining the external and internal forces and by drawing the diagrams of normal, bending and shear forces. Furthermore, the students should learn how to extend the equilibrium analysis to deformable bodies as trusses and beams, how to determine the internal stress distribution on the basis of Saint Venant beam theory, how to draw the elastic curve, how to apply a failure criterion and how to analyze the buckling of columns.

Students should know the kinematics, statics and dynamics of a single point in the framework of the classical (Newtonian) mechanics, vector operations (scalar multiplication, vector addition, dot product, cross product) matrix operations, basic concepts of linear algebra and geometry (analytical and differential). With reference to functions of a single variable, students should know the concepts of limit, derivative, integral, Taylor expansion and solution of linear differential equations having constant coefficients. With reference to functions of many variables, students should know the concepts of partial derivative, multi-dimensional integral and Taylor expansion.

STATICALLY DETERMINATE BEAM SYSTEMS (Theory: 8h and Problems:21.5h): Kinematics and statics of a rigid body; constraints in a plane; ill-disposition of constraints; algebraic study of kinematics; graphical study of kinematics of systems having one degree of freedom (kinematic chain); algebraic study of statics; static-kinematic duality; determination of constraint reactions through auxiliary equations, the Principle of Virtual Work and the graphical method; line of pressure; the characteristics of internal reactions; indefinite equations of equilibrium for plane beams; three-hinged arches; closed-frame structures; Gerber beams. DEFORMABLE BODIES (Theory: 8h and Problems: 4.5h): Analysis of strain; dilatation and shearing strain; the projection of the displacement vector; law of transformation of the strain tensor for rotations of the reference system; principal directions of strain; volumetric dilatation; the tension vector; the stress tensor; the projection of the tension vector; law of transformation of the stress tensor for rotations of the reference system; principal direction of stress; the hydrostatic tensor; the deviatoric tensor; Mohr’s circles; plane stress conditions; indefinite equations of equilibrium; boundary conditions of equivalence; kinematic and static equations written in compact form; static-kinematic duality; the Principle of Virtual Work for deformable bodies. ELASTIC CONSTITUTIVE LAW (Theory: 4h): Elastic potential; linear elasticity; the problem of a linear elastic body; the Lame’ operator; the principle of superposition; Kirchhoff’s theorem; Clapeyron’s theorem; Betti’s reciprocal work theorem; isotropy. STRENGTH CRITERIA (Theory: 3h): Coulomb,Tresca and Von Mises. GEOMETRY OF AREAS (Theory: 4h and Problems: 4.5h): Laws of transformation of the static moment vector and of the moment of inertia tensor; principal axes and moments of inertia; Mohr’s circle; axial symmetry and polar symmetry. THE SAINT VENANT PROBLEM (Theory: 17h and Problems: 16.5h): Fundamental hypotheses; centered axial force; flexure; eccentric axial force and bi-axial flexure; the central core of inertia; energetic orthogonality; torsion in beams of (circular cross section, generic cross section, open and closed thin wall section); shearing loading (center of shear, mean shearing stress according to Jourawski formula, rectangular cross section, mean shearing strain, thin-walled cross section); beam strength analysis; differential equation of the elastic line. CONTINUOUS BEAM AND INFLUENCE LINE (Theory: 3h + Problems: 3h): The influence line drawing is based on first (Betti) and second(Land) reciprocal work theorem. INSTABILITY OF ELASTIC EQUILIBRIUM (Theory: 4h): Discrete mechanical system with one degree of freedom; rectilinear beams with distributed elasticity; kinematic and static boundary conditions. Discrete mechanical system with one degree of freedom; rectilinear beams with distributed elasticity; kinematic and static boundary conditions.

In order to verify the teaching flow, some home-works are assigned.

Delivery modes: Practical activity in the classroom: 20 per cent of total practical activity is done in parallel (i.e. at the same time) by two teachers working in two different classrooms. 80 per cent is done in a single classroom by a single teacher. Practical activity are: Statically determinate beam systems (21.5h): Deformable bodies (4.5h): Geometry of areas (4.5h): The Saint Venant problem (16.5h): Continuous beam and influence line(3h) In the general purposes laboratory on materials and structures: The students, divided in groups of 25 persons, attend tests executed by a technician as follows: Mono-axial compression test on a concrete cube, Mono-axial tensile test on a reinforcement bar, Experimental determination of the shearing center in an aluminum bar having a C cross section and loaded as a cantilever beam. Total time in this laboratory is 1h. In the educational laboratory LADISS: Two equipped magnetic boards are available; the students, divided in groups of 12 persons, are able to execute directly the following tests on them: A simply supported beam subjected to a concentrated and a distributed load. A cantilever beam with the fixed edge subjected to an elastic settlement. A cantilever beam equipped with a displacement transducer able to measure 0.01 mm. The elastic curve of a Gerber beam analyzed through a composition of rotations and displacements. A portal frame having the feet of both colums hinged. Total time in this laboratory is 1h.

Reference books and documents Theory:A.Carpinteri, Structural mechanics: an unified approach, E and FN Spon. Problems:F.P.Beer, E.R.Johnston jr., J.T.DeWolf, D.F.Mazurek Statics and mechanics of materials,Mc Graw Hill Further documents related to the subjects presented will be distributed in the classroom and posted in the official website. In order to show the type of neat and orderly work that students should cultivate in their own solution, same problems solved during written tests are posted on http://www.polito.it/scienza_delle_costruzioni/edili. Further readings R.R. Craig, Mechanics of materials, John Wiley and sons E.Viola, Esercitazioni di Scienza delle Costruzioni, Vol.1, Pitagora Editrice

Lecture slides; Lecture notes; Text book; Practice book; Exercises; Exercise with solutions ; Video lectures (previous years);

Exam: Written test; Compulsory oral exam; Individual essay;

Examination includes two written and one oral tests. Since the focus is on the ability to apply theoretical knowledge to simple professional problems, all books or documents are allowed during written tests. On the contrary, since the focus is on the proven theoretical knowledge, no documents is allowed during oral tests. Written tests are based on problems similar to the problems solved in the classroom. In order to show the type of neat and orderly work that students should cultivate in their own solution, same problems solved during written tests are posted on http://www.polito.it/scienza_delle_costruzioni/edili. The first written test includes three questions:(a)with reference to a statically determinate plane structure it is asked to draw diagrams of characteristics M, N, T (19 scores); (b)in a similar context, it is asked to draw qualitatively an influence line (4 scores); (c)with reference to a predefined area, it is asked to determine some geometrical properties (7 scores). The first test duration is 2.5 hours. The second written test includes two questions:(a)with reference to a Saint Venant solid is asked to determine stresses acting in some predefined positions (20 scores); (b)with reference to a plane stress state, determine the three Mohr’s circles, the principal stresses and a collapse multiplier (10 scores). The second test duration is 2 hours. The oral test is based on two questions drawn from a predefined list. The test duration is 1 hour. Final score is the average value on the three above mentioned tests. In case of doubt, the home-work results can be considered. The maximum score is 30/30 cum laude.

In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.